The Green’s function is a concept developed by George Green in the XIXth century. It is useful to solve inhomogenous differential equations such as :
where the functions f and g are defined on an interval.
More generally, consider an equation with a linear differential operator L :
A Green’s function of a linear operator L is any solution of the equation
where is the Dirac delta-function.
The original equation can be rewritten as follows :
In this way, the Green’s function plays the role of an integral kernel in the solution of the equation .
INHOMOGENEOUS LINEAR EQUATIONS
A first order inhomogenous linear equation is an equation
It can be solved with initial condition :
GREEN’S FUNCTION OF A INHOMOGENOUS LINEAR EQUATION
What is the Green’s function of a first order inhomogenous linear equation ?
The Green’s function satisfies the equation
Then, we apply the solution defined above of a inhomogenous linear equation with the condition
We illustrate the Green’s function on the following equation
The Green’s function is computed as
It is graphically represented at the beginning of the post.
We deduce the solution